Introduction to combinatorial auctions

1. Introduction to combinatorial auctions By Guy Kortsarz

2. A 1-item auction mechanisms • Each bidder submits a bid in an envelope • Auctioneer opens the envelopes, highest bid wins. • The method used usually, called VCG method. • First-price : the one proposing most money. • Second-price discount: But he is given a discount which is only the value of the second highest bidder. • His gain v1-b2

3. On the importance of being truthful • The bidders may cheat • May give prices different than their true values. • A mechanism is Truthful if the bidders say the correct prize. • Interestingly, a truthful mechanism for this simple problem was only designed in 1961 by Vickrey and since then is called: Vickerey Auction.

4. Truthfulness • Let {bi} be the number the bidders bid. • For simplicity say that b1 and b2 contain the first and second highest bid. • Let vibe the true values. Say that b1≥ b2 for the moment (in general we do not assume that). • The net value the first agent gets is:v1-b2 • The bidders may bid higher to try to win (it does not change the payment) • And of course may bidbi<vi

5. On being truthful • Overbidding does not help. Moreover can hurt. Since the agents are rational they will bid truthfuly. • There are three cases depending on v1, b1, and b2. • In any case the assumption is that b1 and b2 are the largest and second largest bid.

6. First case • v1≥b2 • Because we are on the case of overbid,namelyb1>v1 it is clear that agent 1 wins. • If you overbid, it does not make the true value of the item larger than v1. • Never mind what you overbid, you still get v1-b2 money. No point in cheating.

7. Remaining cases • The case that b2>b1 • In this case agent1looses so no matter. • The last case remaining is that v1< b2< b1. • This is a case in which agent 1 looses because of over bidding. • v1-b2<0 • Truthful biding will make him loose but at least does not loose money.

8. But what about underbidding? • b1<v1. Try to save money. • Again let b2 and b1 be the two largest bids. • What if b1<v1<b2? In this case agent 1 does not get the item so truthful or not, same revenue of 0. • If b2<b1. Then the net gain is v1-b2. This is even if the bid is much smaller thanv1.

9. The last case remaining • b1<b2<v1. • Shows that underbidding was a mistake. • 0 net gain versus v1-b2>0. • We call this a dominating strategy. A strategy that is never worse than the value of any other strategy.

10. Many items • The items are U={1,2,….,m} • For now assume that there is aunique copy per items. • We have agents and each agent has a value vi(S), for every subset S U.For every i its 2m values! • The goal is to split sets so agent i gets Siand Si is a disjoint partition of U. • Max I vi(Si) A.K.ASocial welfare.

11. Are values monotone? • Its is natural to assume that if ST then vi(S)≤vi(T). • This assumption is NOT made here. • Very interesting case is SINGLE MINDED BIDDERS. • This means that the bidder wants just one setS and is willing to pay some value for it. But for sets larger than S, his pay is 0! • SMB has very nice theory. But I cal not speak on all subjects.

12. Social welfare: total happiness. • ivi(Si). Note: uses real values. • Now enters the issue of computability of the partition (in economy, did they even care? I don’t know). • This is hard to compute unless we have exponential time. • Discussion of the Independent set problem.

13. An independent set • A collection of pairwise non neighbors

14. An independent set • A collection of pairwise non neighbors

15. Approximating the IS problem • Hastad: no better than n1- approximation (it is much worse actually but the above enough). • This says the following: its as hard to approximate within n1- is as hard as solving exactly! A remarkable result. • This is a result that follows from the famous PCP theorem. • The PCP theorem basically says that SAT has no 1- approximationfor some .

16. First hardness • Hastad showed that getting about n size independent set when there is a size n 1-  independent set is as hard as solving the independent set problem exactly! • Not well known: Berman et al showed that if the PCP theorem holds, there is an  so that IS cannot be approximatedbyn • Amazing: done in 1988, 4 years before the PCP theorem was proved! • Tool: Randomized graph products.

17. Implication for CA • Even if agents are truthful: • Consider one item per edge. • LetVdenote the bidders. • A vertex v ONLY wants its set of edges . • Any solution is an IS. Because single copies. • Say every vertex is willing to pay x for its set but nothing to any other set. • This means that SW roughly sqrt{m}=nNA.

18. Not possible to approximation within roughly sqrt{m}. • Say every v agrees to pay is a x for his (unique) set but nothing for other sets. • The sum of payments is what auction manager wants to maximize. • The auctioneer can only get |I|*x value with |I| the maximum independent set he can compute. • As m can be about n2 this gives roughly sqrt{m} inapproximability.

19. State of the art: in the time written and as far as I know. • Nisan and Mualem: one item many units. • Ratio ½ for single minded bidders. • Improve to FPAS by Briest, Krysta and Voecking. • Dobzinski and Nisan: Not single minded bidders. A ½ ratio with Maximum in Range (MIR) algorithm. Hence polynomial time.

20. More state of the art results • Dobzinski Nisan and Schapira. First to give O(sqrt{m}) randomized truthful algorithm for CA. Note: its for single copy. • See also a paper by Dobzinski in APPROX-RANDOM 2007. • As far as I know, no deterministic algorithm with such results is known. • Big open problem.

21. An interesting recent paper • Krysta and Vocking. On-line algorithm with b≥1 copies per item. • They present anO(m1/(b+1) log(bm)) on-line competitive ratio algorithm. • The algorith is randomized. • But is a distribution over dominating strategies. • Needs exponential power (not a surprise).

22. State of the art continued. • This algorithm asks queries such as: given prices {pi} what is the best partition of the set S of elements. • Note that better than O(m1/(b+1) approximation isNPC (albeit the on-line algorithm uses exponential time procedures). • All on-line algorithm are multiplicative update algorithms. Values to items are multiplied at every round. • Popular items get higher values.

23. Summary • A truthful deterministicsqrt{m} approximation is not known • But is known if we allow randomization. • In randomization the distribution (in my opinion) should be over dominatingstrategies. • Albeit, there are weaker notions (that I don’t like!).

24. Vickery-Clark-Groves Mechanism • We leave pure computer science and enter the sinister subject of economy. • Because we allow the mechanism to set prices for the agents. The problem becomes a non pure computer science problem. • It turns out that without setting prices it is hard to get truthful mechanism.

25. We ignore efficiently issues from now almost till the end. • The best social welfare is the one that divides the sets into {Si}, gives agent i the set Si and among all possiblepartitions,it maximizes the social welfare,namely iv(Si) which is the“total hapiness”. • Our goal is to get a truthful mechanism. • Because of the influence of economy, many times exponential time mechanism are allowed. • In any case we allow any time needed, exponential or beyond.

26. Our goals • High money outcome and truthful. • We set a price pi for agent i. • At the end the net value is v(Si)-pi • Intuitively the price is the damage of the agent to inflicts on other agents. • pj = the maximum social welfare without playerj minus the social welfare the others got (which depends on agentj). • The second term depends on j. • But the first terms does not depend onj.

27. Intuitive explanation • Intuitively the price is the damage of the agent to inflicts on other agents. • The price pj= the maximum social welfare without playerj minus the social welfare the others got (which depends on agentj). • When j participates, it may be that other agent get less value. • Thus we compute the optimum without j for al other agents minus with j.

28. Intuitive explanation • Consider the maximum social welfare without playerj. • Clearly this is at least as large as social welfare the others got when j does participate. Because j participates this second value depends on j. • Clearly the optimization without j is the maximum social revenue the others can get. • Thus the above term is at least 0.

29. In the reverse direction • There are approximation algorithms that use the VCG method. • They define this VCG value on a problem. • They show that under some conditions, there is a good approximation for the problem. Thus the net value VCG is used in approximation algorithm (see some papers by Anupam Gupta and others).

30. The single item mechanism is VCG • Player 1 should pay the optimal social welfare, if it does not participate minusthe social welfare of the others got from the chosen outcome. • Note: if 1 does not participate the social welfare is v2=b2 because of truthfulness. • SW for others when agent 1 participates 0. • b2-0=0. This is indeed the price for player 1. • Thus b2 price indeeda VCG mechanism. • Net gain v1-b2

31. This mechanism is truthful (even if hard to calculate) • pj = the maximum social welfare without playerj minus the social welfare the others got. • The first term does not depend on agent j. • So, think of this term as 0. In such a case the value minus discount is vi(Si)+the social welfare the other agents got. • Which is what is maximized. Hence cheating will be a mistake.

32. Why is the term not related to i is inserted? • Player i should pay the optimal social welfare, if it does not participate minusthe social welfare of the others got from chosen outcome. • This number is clearly at least 0, hence no negative pays. • Also the same reasoning shows that vi-pi is always at least 0. No loss.

33. Approximation: the missing link • If you want to maximize the social welfare or social revenue, it may be NP-hard,to do so. • Study special cases for example. Special welfare functions for example. • It comes natural to CS people to say: approximate the social welfare/revenue. • We shall see later that approximation can kill truthfulness.

34. Approximation: the missing link • A quote by Woody Allen: • When I was kidnap my parents took immediate steps.

35. Approximation: the missing link • A quote by Woody Allen: • When I was kidnap my parents took immediate steps. • They rented my room!

36. In CS conferences • Here and there people from Economy attend our conferences. • They say: we only care on optimum. • They say: worst case is not a good measure. • They say: your approximation ratios are not practical. • Hence we took immediate steps!

37. In CS conferences • Here and there people from Economy attend our conferences. • They say: we only care on optimum. • They say: worst case is not a good measure. • They say: your approximation ratios are not practical. • Hence we took immediate steps! • We said that we don’t care.

38. I wonder • Is it true that algorithmic mechanism designset minuswhat they study in economy is simply approximation algorithms? • Approximation algorithm have all what people in economy may not like. • For example worse case. In Economy is with respect to distributions. • Get a less than optimal (approximation) social welfare. In economy I think its either optimal or not. Note interestes in approximation it seems.

39. Approximation killstruthfulness! • Example: consider two agents and two items. Both evaluate each item separately by 1.9 and both items by 3.1. • Optimum to maximize social welfare : give each, one item. Total welfare 3.8. • The VCG payment: 3.1-1.9=1.2. • Consider truncating values down Appr’. • If they bid truthfully the approximation algorithm will give values (1,3).

40. No good deed goes unpunished • With values (1,3) its better to give both items to one of them and get social welfare 3. • The VCG payment of the one who got the items: 3.1 (we use real values for the payment. But using rounded ones fails as well!). • Net revenue is 0.

41. Cheating is better • Say that exactly one of the players cheats. • Says: for any subset (except the empty set), I am wiling to pay 3. • Giving each one a single item getsrevenue 4,clearly, best possible. • The non truthful is charged: 3.1-1.9=2.1. • Net revenue 0.9. Non truthful mechanism.

42. Approximation lost us VCG, what to do? • Well, try to find mechanism that will give a good approximation and still are truthful. • Example: say that there is one item in m copies. • Every agent has for every k≤m a value vi(k). It is assumed that vi(k)≤ vi(k+1). • A mechanism has to allocate miunits to agent i so that mi=m and maximize vi(mi)

43. Solving it even with full knowledge is NPC: Knapsack • An example of a modern result. • Due to Dobzinski and Nisan. • There exists a truthful mechanism that gets at least ½ the optimal social welfare. • This is called ratio 2 at times. The hard part is making it truthful. • The algorithm is efficient as it is maximum in (small) range. You search for the best among few solutions. • Best possible of his type.

44. Summary • Auctions just one important example. • Voting is another example. • Also extensively studied is price of anarchy and price of stability. Maybe to be described in a future lecture. • Also extensively studied: how hard is it to compute a Nash Equilibrium. This has a complete class of problems already. Thus hardness results. • Leaving cynicism aside: the relation between the community of Algorithmic game theory and economy should be made much closer in my opinion. • To begin with many (but many) results rediscovered in CS. Known from the 1950’s in Economy!